**From:** Stuart Armstrong (*dragondreaming@googlemail.com*)

**Date:** Sat Mar 22 2008 - 03:43:45 MDT

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Some extra precisions on my long lost example (long lost, since it

happened two long weeks ago...)

*> > The answer, whatever it is, will have simpler, quasi answers -
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*>
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*> > incomplete but informative. The hash function equivalents
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*>
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*> > will not be simpler than the hash function equivalent to the full
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*> > question.
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Examples of simpler questions: for the derivative, we have interim

results: the derivative of a monomial is a monomial, the derivative of

a polynomial of n-th degree is a polynomial of n-1 th degree. These

are simpler than knowing the full derivative rule, but to list them

under the hash function is actualy more complicated than listing the

GLUT for the derivative itself (as we first have to define the subsets

"monomials" and "polnomial of n-th degree").

*> > 2) The implicit infinity.
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*> > Implicit in the definition of differentiation is the fact that we
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*> > could differentiate any polynomial (with, say, rational coefficients).
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*> > The definition of differentiation to f(S) does not extend to infinity
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*> > in this way; in fact, there is no evident extention of f(S).
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*> What? Even if you have infinitely many polynomials, each has a
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*> derivative, the only difference being that there is no "top dog".
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*> Strategically, would it have been better to stick with a crippled
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*> definition of derivative, oh, something along the lines of either
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*> "we are not going to consider S to be infinite", or "a qDerivative
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*> is a derivative of a polynomial function, but the domain of
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*> qDerivative is by definition finite"? Or something like that?
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The derivative can be defined for any polynomial, and can be defined

in a finite (and very short) sentence. A finite definition, generating

an infinite number of relations.

Now try and go the other way; from a GLUT, try and deduce the general

rule. If we have the GLUT in polynomial form, the universal rule is

easy to deduce; if we have the GLUT in some hash function equivalent,

we can't do so.

*> I think that the whole infinity digression is a red-herring to your otherwise
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*> brilliant analysis (and I don't often call what other posters do "brilliant"),
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*> reserving that appellation strictly to apply to my own missives only.
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I'm actually blushing there :-) Many thanks. However, the red herring

is still, in my view, a distinction between a GLUT and a

hash-equivalent GLUT.

Stuart

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